#350428

order	512
nilpotency class	3
isoclinism	isoclinic class of #346398
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1092 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2)(3,6)(4,5)(7,8)(9,10,12,15)(11,19,20,24)(13,18,14,17)(16,22,23,21)(25,26,29,32)(27,30,34,37)(28,31,35,38)(33,39,40,36);(2,5)(6,8)(10,15)(16,23)(17,18)(19,24)(28,35)(31,38)(33,40)(36,39);(1,3,4,7)(2,6,5,8)(9,11)(10,16)(12,20)(13,22)(14,21)(15,23)(17,24)(18,19)(25,27)(26,30)(28,33)(29,34)(31,36)(32,37)(35,40)(38,39);(9,11)(10,16)(12,20)(13,21)(14,22)(15,23)(17,19)(18,24)(25,28)(26,31)(27,33)(29,35)(30,36)(32,38)(34,40)(37,39)

Polycyclic presentation:

2^9|a^2=hi,c^2=eh,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=dg,f^a=fh,g^a=gh,d^b=di,f^c=fh,g^c=gi,f^d=fi,g^d=gh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 257624);